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In this paper, we establish a Kantorovich duality for weak optimal totalvariation transport problems. As consequences, we recover a version of dualityformula for partial optimal transports established by Caffarelli and McCann;and we also get another proof of Kantorovich-Rubinstein Theorem for generalizedWasserstein distance $\widetilde{W}_1^{a,b}$ proved before by Piccoli andRossi. Then we apply our duality formula to study generalized Wassersteinbarycenters. We show the existence of these barycenters for measures withcompact supports. Finally, we prove the consistency of our barycenters.

Weak optimal total variation transport problems and generalized Wasserstein barycenters